Minority-spin $t_{2g}$ states and the degree of spin polarization in ferromagnetic metallic La$_{2-2x}$Sr$_{1+2x}$Mn$_2$O$_7$ ($x=0.38$)

Using angle-resolved photoemission spectroscopy (ARPES), we investigate the electronic band structure and Fermi surface of ferromagnetic La$_{2-2x}$Sr$_{1+2x}$Mn$_2$O$_7$ ($x=0.38$). Besides the expected two hole pockets and one electron pocket of majority-spin $e_g$ electrons, we show an extra electron pocket around the $\Gamma$ point. A comparison with first-principles spin-polarized band-structure calculations shows that the extra electron pocket arises from $t_{2g}$ electrons of minority-spin character, indicating this compound is not a complete half-metallic ferromagnet, with similar expectations for lightly-doped cubic manganites. However, our data suggest that a complete half-metallic state is likely to be reached as long as the bandwidth is mildly reduced. Moreover, the band-resolved capability of ARPES enables us to investigate the band structure effects on spin polarization for different experimental conditions.